超几何-Meijer逼近子的高阶参数化
High-Order Parametrization of the Hypergeometric-Meijer Approximants
AI总结：
本文将超几何-Meijer逼近子参数化算法扩展到任意高阶，并引入非微扰信息加速收敛，在非谐振子基态能量和SQ晶格磁化率等问题上取得了精确结果。
摘要：
在之前的文章中，我们证明了基于大阶渐近行为，可以通过特定超几何逼近子的参数化来近似一个发散级数。解析延拓随后通过超几何逼近子的Mellin-Barnes积分表示进行，或者等价地使用Meijer G-函数的等价形式。参数化过程涉及求解一组非线性耦合方程，使用普通个人计算机在高阶情况下很难实现（甚至可能无法实现）。在这项工作中，我们将近似算法扩展为能够在短时间内适应给定级数的任意阶（高阶或低阶）。该扩展还允许我们利用非微扰信息，例如强耦合和大阶渐近数据，这些信息总是被用来加速收敛。我们将该算法应用于$x^4$非谐振子基态能量的不同阶（最高到O($29$)），包括使用和不使用非微扰信息的情况。我们还考虑了$\mathcal{PT}$对称的$ix^{3}$非谐振子基态能量的可用$20$阶，以及其强耦合展开的给定$20$阶，即等价地Yang-Lee模型。对于高阶弱耦合参数化，我们获得了基态能量以及描述强耦合和大阶渐近行为的非微扰参数的精确结果。非微扰数据的使用非常明显地加速了收敛。我们还考虑了$SQ$晶格中磁化率的高温展开，并得到了临界指数和临界温度的精确预测。
英文摘要：
In previous articles, we showed that, based on large-order asymptotic behavior, one can approximate a divergent series via the parametrization of a specific hypergeometric approximant. The analytical continuation is then carried out through a Mellin-Barnes integral representation of the hypergeometric approximant or equivalently using an equivalent form of the Meijer G-Function. The parametrization process involves the solution of a non-linear set of coupled equations which is hard to achieve (might be impossible) for high orders using normal PCs. In this work, we extend the approximation algorithm to accommodate any order (high or low) of the given series in a short time. The extension also allows us to employ non-perturbative information like strong-coupling and large-order asymptotic data which are always used to accelerate the convergence. We applied the algorithm for different orders (up to O($29$)) of the ground state energy of the $x^4$ anharmonic oscillator with and without the non-perturbative information. We also considered the available $20$ orders for the ground sate energy of the $\mathcal{PT}-$symmetric $ix^{3}$ anharmonic oscillator as well as the given $20$ orders of its strong-coupling expansion or equivalently the Yang-Lee model. For high order weak-coupling parametrization, accurate results have been obtained for the ground state energy and the non-perturbative parameters describing strong-coupling and large-order asymptotic behaviors. The employment of the non-perturbative data accelerated the convergence very clearly. The High temperature expansion for the susceptibility within the $SQ$ lattice has been also considered and led to accurate prediction for the critical exponent and critical temperature.
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